Octave 3.8, jcobi/1

Average Error: 16.2 → 6.1

Time: 5.3s

Precision: 64

\[\alpha \gt -1 \land \beta \gt -1\]

Math

\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \le 37342693843744496:\\ \;\;\;\;\frac{e^{\log \left(\mathsf{fma}\left(\beta, \frac{1}{\left(\alpha + \beta\right) + 2}, -\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}\right) \cdot \frac{\sqrt[3]{\beta}}{\left(\alpha + \beta\right) + 2} - \mathsf{fma}\left(4, \frac{1}{{\alpha}^{2}}, -\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]

C

double code(double alpha, double beta) {
	return ((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0);
}

↓

double code(double alpha, double beta) {
	double temp;
	if ((alpha <= 3.73426938437445e+16)) {
		temp = (exp(log(fma(beta, (1.0 / ((alpha + beta) + 2.0)), -((alpha / ((alpha + beta) + 2.0)) - 1.0)))) / 2.0);
	} else {
		temp = ((((cbrt(beta) * cbrt(beta)) * (cbrt(beta) / ((alpha + beta) + 2.0))) - fma(4.0, (1.0 / pow(alpha, 2.0)), -fma(2.0, (1.0 / alpha), (8.0 * (1.0 / pow(alpha, 3.0)))))) / 2.0);
	}
	return temp;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

1. Split input into 2 regimes

2. if alpha < 3.73426938437445e+16

   1. Initial program 0.3

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
   2. Using strategy rm

   3. Applied div-sub 0.3

      \[\leadsto \frac{\color{blue}{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2} - \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)} + 1}{2}\]

   4. Applied associate-+l- 0.3

      \[\leadsto \frac{\color{blue}{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}}{2}\]
   5. Using strategy rm

   6. Applied div-inv 0.3

      \[\leadsto \frac{\color{blue}{\beta \cdot \frac{1}{\left(\alpha + \beta\right) + 2}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\]

   7. Applied fma-neg 0.3

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\beta, \frac{1}{\left(\alpha + \beta\right) + 2}, -\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right)}}{2}\]
   8. Using strategy rm

   9. Applied add-exp-log 0.3

      \[\leadsto \frac{\color{blue}{e^{\log \left(\mathsf{fma}\left(\beta, \frac{1}{\left(\alpha + \beta\right) + 2}, -\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right)\right)}}}{2}\]

   if 3.73426938437445e+16 < alpha

   1. Initial program 49.7

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
   2. Using strategy rm

   3. Applied div-sub 49.7

      \[\leadsto \frac{\color{blue}{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2} - \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)} + 1}{2}\]

   4. Applied associate-+l- 48.1

      \[\leadsto \frac{\color{blue}{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}}{2}\]
   5. Using strategy rm

   6. Applied *-un-lft-identity 48.1

      \[\leadsto \frac{\frac{\beta}{\color{blue}{1 \cdot \left(\left(\alpha + \beta\right) + 2\right)}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\]

   7. Applied add-cube-cbrt 48.2

      \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}\right) \cdot \sqrt[3]{\beta}}}{1 \cdot \left(\left(\alpha + \beta\right) + 2\right)} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\]

   8. Applied times-frac 48.2

      \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{1} \cdot \frac{\sqrt[3]{\beta}}{\left(\alpha + \beta\right) + 2}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\]

   9. Simplified 48.2

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}\right)} \cdot \frac{\sqrt[3]{\beta}}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\]

   10. Taylor expanded around inf 18.4

       \[\leadsto \frac{\left(\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}\right) \cdot \frac{\sqrt[3]{\beta}}{\left(\alpha + \beta\right) + 2} - \color{blue}{\left(4 \cdot \frac{1}{{\alpha}^{2}} - \left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}}{2}\]

   11. Simplified 18.4

       \[\leadsto \frac{\left(\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}\right) \cdot \frac{\sqrt[3]{\beta}}{\left(\alpha + \beta\right) + 2} - \color{blue}{\mathsf{fma}\left(4, \frac{1}{{\alpha}^{2}}, -\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}}{2}\]
3. Recombined 2 regimes into one program.

4. Final simplification 6.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 37342693843744496:\\ \;\;\;\;\frac{e^{\log \left(\mathsf{fma}\left(\beta, \frac{1}{\left(\alpha + \beta\right) + 2}, -\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}\right) \cdot \frac{\sqrt[3]{\beta}}{\left(\alpha + \beta\right) + 2} - \mathsf{fma}\left(4, \frac{1}{{\alpha}^{2}}, -\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020053 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2)) 1) 2))